Short answer: Copper monolithic bullets aren't lead, so don't expect it will do things the same as lead.
Long answer below.
I changed to using copper bullets for the same exact reason. I do not know what bullet you are using, there are lots of copper bullet manufacturers currently providing lathe turned solids, or lathe turned HP.
I use the GSC bullets, or those from Barnes. I hand load everything I shoot.
http://www.gscustomusa.com/sp.html
If you notice that is not a HP bullet, yet it still expanded as you can see above. Velocity was about 3780fps at the muzzle. Velocity alone can cause even a solid copper bullet to perform. If you are still having troubles, change your bullet. These are also fine for punching paper at longer ranges.
http://www.gscustomusa.com/a.html
If you are looking for expansion for a 308 at 150yds or more, you should consider this:
https://www.wholesalehunter.com/Product/Details/1007346
https://www.midwayusa.com/product/10...free-box-of-50
Change your bullet, your approach to hunting, your approach to shot placement or perhaps a combination of any or all of the above. I have been using a GSC SP bullet for all my deer hunting, in Texas ANY centerfire is legal, so I use .223"Rem, .172" VHA or a .172"FB with all copper bullets. I shoot for the head and more specifically for the brain stem. Either I get close enough to do this while stalking, or I get a steady enough rest to place my shot. Heart shots are also sometimes used and they are "different " than the response I would get from a lead core bullet, I don't get lead fragments.
At velocities higher than 4000fps the monometal solid SP created plenty of hydro static shock for either a heart/lung shot, or anything else, but the monometal bullet ensures that I do not find myself crunching lead in a steak.
If you are using these and not getting the results you want, then get closer.
Also, if you are able to hit the top of the heart from a front quartering shot of about 30° or more from either level or up or down angle, you are likely going to hit the off side lung. That usually will terminate a deer in my experience because the hydrostatic shock impacting the spine directly above the heart receiving trauma.